Obstructions to branch-decomposition of matroids

A (δ,γ)-net in a matroid M is a pair (N,P) where N is a minor of M, P is a set of series classes in N, |P|⩾δ, and the pairwise connectivity, in M, between any two members of P is at least γ. We prove that, for any finite field F, nets provide a qualitative characterization for branch-width in the class of F-representable matroids. That is, for an F-representable matroid M, we prove that: (1) if M contains a (δ,γ)-net where δ and γ are both very large, then M has large branch-width, and, conversely, (2) if the branch-width of M is very large, then M or M∗ contains a (δ,γ)-net where δ and γ are both large.